Real thermodynamics is celebrated for its precision, power,
generality, and elegance. However, all too often, students are taught
some sort of pseudo-thermodynamics that is infamously confusing, lame,
restricted, and ugly. This document is an attempt to do better, i.e.
to present the main ideas in a clean, simple, modern way.

The first law of thermodynamics is usually stated in a very
unwise form.

We will see how to remedy this.

The second law is usually stated in a very unwise form.

We
will see how to remedy this, too.

The so-called third law is a complete loser. It is beyond
repair.

We will see that we can live without it just fine.

Many of the basic concepts and terminology (including heat,
work, adiabatic, etc.) are usually given multiple
mutually-inconsistent definitions.

We will see how to avoid the
inconsistencies.

Many people remember the conventional “laws” of thermodynamics by
reference to the following joke:1

0) You have to play the game;

1) You can’t win;

2) You can’t break even, except on a very cold day; and

3) It doesn’t get that cold.

It is not optimal to formulate thermodynamics in terms of a short list
of enumerated laws, but if you insist on having such a list, here it
is, modernized and clarified as much as possible. The laws appear in
the left column, and some comments appear in the right column:

The zeroth law of thermodynamics tries to tell us that certain
thermodynamical notions such as “temperature”, “equilibrium”,
and “macroscopic state” make sense.

Sometimes these make sense,
to a useful approximation … but not always. See chapter 3.

The first law of thermodynamics states that energy obeys a local
conservation law.

The conventional
so-called third law alleges that the entropy of some things goes to
zero as temperature goes to zero. This is never true, except perhaps
in a few extraordinary, carefully-engineered situations. It is never
important. See chapter 4.

To summarize the situation, we have two laws (#1 and #2) that are
very powerful, reliable, and important (but often misstated and/or
conflated with other notions) plus a grab-bag of many lesser laws that
may or may not be important and indeed are not always true (although
sometimes you can make them true by suitable engineering). What’s
worse, there are many essential ideas that are not even hinted at in
the aforementioned list, as discussed in chapter 5.

We will not confine our discussion to some small number of axiomatic
“laws”. We will carefully formulate a first law and a second law,
but will leave numerous other ideas un-numbered. The rationale for
this is discussed in section 6.9.

The relationship of thermodynamics to other fields is indicated in
figure 0.1. Mechanics and many other fields use the
concept of energy without worrying very much about entropy.
Meanwhile, information theory and many other fields use the concept of
entropy without worrying very much about energy; for more on this see
chapter 21. The hallmark of thermodynamics is that it
uses both energy and entropy.

This section is meant to provide an overview. It mentions the main
ideas, leaving the explanations and the details for later. If you
want to go directly to the actual explanations, feel free to skip this
section.

There is an important distinction between fallacy and
absurdity. An idea that makes wrong predictions every time is
absurd, and is not dangerous, because nobody will pay any attention
to it. The most dangerous ideas are the ones that are often correct
or nearly correct, but then betray you at some critical moment.

Most of the fallacies you see in thermo books are pernicious
precisely because they are not absurd. They work OK
some of the time, especially in simple “textbook” situations
… but alas they do not work in general.

The main goal here is to formulate the subject in a way that is less
restricted and less deceptive. This makes it vastly more reliable in
real-world situations, and forms a foundation for further learning.

In some cases, key ideas can be reformulated so that they work just
as well – and just as easily – in simple situations, while
working vastly better in more-general situations. In the few
remaining cases, we must be content with less-than-general results,
but we will make them less deceptive by clarifying their limits of
validity.

We distinguish cramped thermodynamics from uncramped
thermodynamics as shown in figure 0.2.

On the left side of the diagram, the system is constrained to
move along the red path, so that there is only one way to get from
A to Z.

In contrast, on the right side of the diagram, the
system can follow any path in the (S,T) plane, so there are
infinitely many ways of getting from A to Z, including the simple
path A→Z along a contour of constant entropy, as well
as more complex paths such as A→Y→Z and
A→X→Y→Z. See chapter 18
for more on this.

Indeed, there are infinitely many paths from A back to
A, such as A→Y→Z→A and
A→X→Y→Z→A. Paths
that loop back on themselves like this are called thermodynamic
cycles. Such a path returns the system to its original state, but
generally does not return the surroundings to their original state.
This allows us to build heat engines, which take energy from a heat
bath and convert it to mechanical work.

There are some simple ideas such as specific heat
capacity (or molar heat capacity) that can be developed within the
limits of cramped thermodynamics, at the high-school level or even
the pre-high-school level, and then extended to all of
thermodynamics.

Alas there are some other ideas such as “heat
content” aka “thermal energy content” that seem attractive in the
context of cramped thermodynamics but are extremely deceptive if you
try to extend them to uncramped situations.

Even when cramped ideas (such as heat capacity) can be extended, the
extension must be done carefully, as you can see from the fact that
the energy capacity CV is different from the enthalpy capacity
CP, yet both are widely (if not wisely) called the “heat”
capacity.

Uncramped thermodynamics has a certain irreducible
amount of complexity. If you try to simplify it too much, you
trivialize the whole subject, and you arrive at a result that wasn’t
worth the trouble. When non-experts try to simplify the subject, they
all-too-often throw the baby out with the bathwater.

You can’t do thermodynamics without entropy. Entropy is
defined in terms of statistics. As discussed in chapter 2,
people who have some grasp of basic probability can understand
entropy; those who don’t, can’t. This is part of the price of
admission. If you need to brush up on probability, sooner is better
than later. A discussion of the basic principles, from a modern
viewpoint, can be found in reference 1.

We do not define entropy in terms of energy, nor vice versa. We do
not define either of them in terms of temperature. Entropy and
energy are well defined even in situations where the temperature is
unknown, undefinable, irrelevant, or zero.

Uncramped thermodynamics is intrinsically multi-dimensional.
Even the highly simplified expression dE = − PdV + TdS
involves five variables. To make sense of this requires partial
derivatives. If you don’t understand how partial derivatives work,
you’re not going to get very far.

Furthermore, when using partial derivatives, we must not assume that
“variables not mentioned are held constant”. That idea is a dirty
trick than may work OK in some simple “textbook” situations, but
causes chaos when applied to uncramped thermodynamics, even when
applied to something as simple as the ideal gas law, as discussed in
reference 2. The fundamental problem is that the various
variables are not mutually orthogonal. Indeed, we cannot even define
what “orthogonal” should mean, because in thermodynamic
parameter-space there is no notion of angle and not much notion of
length or distance. In other words, there is topology but no
geometry, as discussed in section 7.7. This is another
reason why thermodynamics is intrinsically and irreducibly
complicated.

Uncramped thermodynamics is particularly intolerant of sloppiness,
partly because it is so multi-dimensional, and partly because there
is no notion of orthogonality. Unfortunately, some thermo books are
sloppy in the places where sloppiness is least tolerable.

Some fraction of this mess can be cleaned up just by being careful
and not taking shortcuts. Also it may help to visualize
partial derivatives using the methods presented in
reference 3. Even more of the mess can be
cleaned up using differential forms, i.e. exterior derivatives and
such, as discussed in reference 4. This raises the
price of admission somewhat, but not by much, and it’s worth it.
Some expressions that seem mysterious in the usual textbook
presentation become obviously correct, easy to interpret, and indeed
easy to visualize when re-interpreted in terms of gradient vectors.
On the other edge of the same sword, some other mysterious
expressions are easily seen to be unreliable and highly deceptive.

You must appreciate the fact that not every vector field is the
gradient of some potential. Many things that non-experts wish
were gradients are not gradients. You must get your head around this
before proceeding. Study Escher’s “Waterfall” as discussed in
reference 4 until you understand that the water there
has no well-defined height. Even more to the point, study the RHS of
figure 7.4 until you understand that there is no well-defined
height function, i.e. no well-defined Q as a function of state.
See also section 7.2.

The term “inexact differential” is sometimes used in this
connection, but that term is a misnomer, or at best a highly
misleading idiom. We prefer the term ungrady one-form. In any
case, whenever you encounter a path-dependent integral, you must keep in
mind that it is not a potential, i.e. not a function of state. See
chapter 18 for more on this.

To say the same thing another way, we will not express the first law
as dE = dW + dQ or anything like that, even though it is
traditional in some quarters to do so. For starters, although such
an equation may be meaningful within the narrow context of cramped
thermodynamics, it is provably not meaningful for uncramped
thermodynamics, as discussed in section 7.2 and
chapter 18. It is provably impossible for there to be any
W and/or Q that satisfy such an equation when thermodynamic
cycles are involved.

Even in cramped situations where it might be possible to split E
(and/or dE) into a thermal part and a non-thermal part, it is
often unnecessary to do so. Often it works just as well (or better!)
to use the unsplit energy, making a direct appeal to the conservation
law, equation 1.2.

Almost every newcomer to the field tries to apply
ideas of “thermal energy” or “heat content” to uncramped
situations. It always almost works ... but it never
really works. See chapter 18 for more on this.

On the basis of history and etymology, you might
think thermodynamics is all about heat, but it’s not. Not anymore.
By way of analogy, there was a time when what we now call
thermodynamics was all about phlogiston, but it’s not anymore.
People wised up. They discovered that one old, imprecise idea
(phlogiston) could be and should be replaced two new, precise ideas
(oxygen and energy). More recently, it has been discovered that one
old, imprecise idea (heat) can be and should be replaced by two new,
precise ideas (energy and entropy).

Heat remains central to unsophisticated cramped thermodynamics, but
the modern approach to uncramped thermodynamics focuses more on
energy and entropy. Energy and entropy are always well defined, even
in cases where heat is not.

The idea of entropy is useful in a wide range of situations,
some of which do not involve heat or temperature.
As shown in figure 0.1, mechanics
involves energy, information theory involves entropy,
and thermodynamics involves both energy and entropy.

You can’t do thermodynamics without energy and entropy.

There are multiple mutually-inconsistent definitions of “heat” that
are widely used – or you might say wildly used – as discussed in
section 16.1. (This is markedly different from the situation
with, say, entropy, where there is really only one idea, even if this
one idea has multiple corollaries and applications.) There is no
consensus as to “the” definition of heat, and no prospect of
achieving consensus anytime soon. There is no need to achieve
consensus about “heat”, because we already have consensus about
entropy and energy, and that suffices quite nicely. Asking students
to recite “the” definition of heat is worse than useless; it
rewards rote regurgitation and punishes actual understanding of the
subject.

Our thermodynamics applies to systems of any
size, large or small ... not just large systems. This is important,
because we don’t want the existence of small systems to create
exceptions to the fundamental laws. When we talk about the entropy
of a single spin, we are necessarily thinking in terms of an
ensemble of systems, identically prepared, with one spin per
system. The fact that the ensemble is large does not mean that the
system itself is large.

Even in special situations where the
notion of “thermal energy” is well defined, we do not pretend that
all thermal energy is kinetic; we recognize that random potential
energy is important also. See section 8.3.3.